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The Lipschitz Uniqueness Theorem for Solutions to IVPs on First Order ODEs

Recall that a continuous function $f \in C(D, \mathbb{R})$ is said to satisfy a Lipschitz condition on $D$ if there exists an $L > 0$ such that for all $(t, x), (t, y) \in D$ we have that:

(1)

\begin{align} \quad | f(t, x) - f(t, y) | \leq L |x - y| \end{align}

We now prove an important result for the uniqueness of solutions to the IVP $x' = f(t, x)$ with $x(\tau) = \xi$, $f \in C(D, \mathbb{R})$ which says that if $f$ satisfies a Lipschitz condition on $D$ then any solution $\phi$ to this IVP is unique on any interval $[\tau - d, \tau + d]$.